Slide 1PROGRAMME 8
Equation of a straight line
Tangents and normals to a curve at a given point
Curvature
STROUD
Programme 8: Differentiation applications 1
Equation of a straight line
Tangents and normals to a curve at a given point
Curvature
Equation of a straight line
Programme 8: Differentiation applications 1
The basic equation of a straight line is:
where:
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Equation of a straight line
Programme 8: Differentiation applications 1
Given the gradient m of a straight line and one point (x1, y1) through which it passes, the equation can be used in the form:
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Equation of a straight line
Programme 8: Differentiation applications 1
If the gradient of a straight line is m and the gradient of a second straight line is m1 where the two lines are mutually perpendicular then:
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Programme 8: Differentiation applications 1
Equation of a straight line
Tangents and normals to a curve at a given point
Curvature
Tangents and normals to a curve at a given point
Tangent
Programme 8: Differentiation applications 1
The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point:
The equation of the tangent can then be found from the equation:
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Tangents and normals to a curve at a given point
Normal
Programme 8: Differentiation applications 1
The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point:
The equation of the normal (perpendicular to the tangent) can then be found from the equation:
STROUD
Programme 8: Differentiation applications 1
Equation of a straight line
Tangents and normals to a curve at a given point
Curvature
Curvature
Programme 8: Differentiation applications 1
The curvature of a curve at a point on the curve is concerned with how quickly the curve is changing direction in the neighbourhood of that point.
Given the gradients of the curve at two adjacent points P and Q it is possible to calculate the change in direction = 2 - 1
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Curvature
Programme 8: Differentiation applications 1
The distance over which this change takes place is given by the arc PQ.
For small changes in direction and small arc distance s the average rate of change of direction with respect to arc length is then:
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Curvature
Programme 8: Differentiation applications 1
A small arc PQ approximates to the arc of a circle of radius R where:
So
and in the limit as
Which is the curvature at P; R being the radius of curvature
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Curvature
Programme 8: Differentiation applications 1
The radius of curvature R can be shown to be given by:
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Programme 8: Differentiation applications 1
Equation of a straight line
Tangents and normals to a curve at a given point
Curvature
Centre of curvature
Programme 8: Differentiation applications 1
If the centre of curvature C is located at the point (h, k) then:
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Learning outcomes
Recognize the relationship satisfied by two mutually perpendicular lines
Derive the equations of a tangent and a normal to a curve
Evaluate the curvature and radius of curvature at a point on a curve
Locate the centre of curvature for a point on a curve
Programme 8: Differentiation applications 1
y
mxc